arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1012.3918math.CO
收藏

大的无B_d和无并子族

Large B_d-free and union-free subfamilies

János Barát, Zoltán Füredi, Ida Kantor, Younjin Kim, Balázs Patkós

备注：
8 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

对于属性$Γ$和集合族$\cF$，令$f(\cF,Γ)$为具有属性$Γ$的$\cF$的最大子族的大小。对于正整数$m$，令$f(m,Γ)$为所有$m$族中$f(\cF,Γ)$的最小值。如果家族$\cF$没有$2^d$不同集合的子家族$\cF'=\{F_I: I \subseteq [d]\}$，使得对于每个$I,J \subseteq [d]$，$F_I \cup F_J=F_{I \cup J}$和$F_I \cap F_J = F_{I \cap J}$都成立，则该家族被称为无$B_d$.如果$F_1,..,F_{a+1}$在$\FF$中是不同的集合，则族$\cF$是$a$ -union free的，如果$F_1\cup ... F_a \neq F_{a+1}$。我们验证了Erd \H os和Shelah关于$f(m, B_2\text{\rm -free})=Θ(m^{2/3})$的猜想。我们还得到了$f(m, B_d\text{\rm -free})$和$f(m,a\text{\rm -union free})$的上下界。

英文摘要：

For a property $Γ$ and a family of sets $\cF$, let $f(\cF,Γ)$ be the size of the largest subfamily of $\cF$ having property $Γ$. For a positive integer $m$, let $f(m,Γ)$ be the minimum of $f(\cF,Γ)$ over all families of size $m$. A family $\cF$ is said to be $B_d$-free if it has no subfamily $\cF'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold. A family $\cF$ is $a$-union free if $F_1\cup ... F_a \neq F_{a+1}$ whenever $F_1,..,F_{a+1}$ are distinct sets in $\FF$. We verify a conjecture of Erd\H os and Shelah that $f(m, B_2\text{\rm -free})=Θ(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union free})$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑